Page 237 - J. C. Turner "History and Science of Knots"

228 History and Science of Knots

These yield the following presentation for the fundamental group of the
Trefoil Knot:

(C1, C2, C3, C4 : C1 C2 C3 1 = C1 C3 1 C4 = C2 C4C3 1 = 1)
In fact, this is not the most economical presentation possible: using Tietze
operations we can reduce the number of generators to 2, and the number of
relations to 1, thus:

(Cl, C2 : C1C2C1 = C2C1C2)
Wirtinger's presentation is derived in similar fashion , but with generators
being associated with overpass arcs, rather than with regions (see [25] for
details). The end-result is, of course, the same.
Further contributions by Dehn are described in the next section on Alexan-
der's work.

8. James Alexander 's Influence

Applications of the fundamental group quickly yielded several breakthroughs.
Proofs of the existence of non-trivial knots, via knot groups, had already been
given by Tietze as early as 1906. However, the first successes from use of the
knot group lay 'in the verification of the correctness of the knot tables. To
achieve this, the group was used with other tools which Henri Poincare and
Enrico Betti had introduced. The proof that Betti numbers and torsion coef-
ficients define combinatorial knot-invariants was first given by James Waddel
Alexander (1888-1971), a professor of mathematics at Princeton University
and later at the Institute for Advanced Studies. Collaborating with G. B.
Briggs and using the torsion numbers, he distinguished all tabled knots up to
8 crossings and all, except three pairs, up to 9 crossings [7]. Alexander also
showed that two 3-dimensional manifolds may have the same Betti numbers,
torsion coefficients and fundamental group and yet not be homeomorphic. His
example, of course, involved knot complements. Thus he had shown that a
knot contains (at least a priori) more information than just its group.
With the tools just introduced, Dehn proved that an arbitrary knot K, its
mirror image K*, and its version with reversed orientation K, produce three
knot-complement groups which are mutually isomorphic. (Later Reidemeister
also proved this, more rigorously.) Using ir1 to denote a knot-complement
group, this theorem is stated as follows:
ir1(R3 ir1(R3
- K, p) - ir1(R'3 - K*, p) - - K, p)
It was thus realized that the complement alone could not provide complete
invariants. The situation was repaired by equipping the complement with an